ABSTRACT
The Hamming codes can be extended to a SEC-DED code by adding one additional parity-check bit. These codes, known as extended Hamming codes, use an extra parity bit to increase the Hamming distance between any two valid encodings. Recall that the Hamming distance between two strings of equal length is the number of bits at which the corresponding symbols differ (see §3.3.1). For example, the 8-bit strings 01010101 and 10101010 have a Hamming distance of 8 because all eight bits differ, but 10001000 and 10001001 have a Hamming distance of 1 because only the lsb differs. Encoding values with a Hamming distance greater than 1 require more bits to represent fewer unique values than the minimum 2n values with n bits that the standard binary encoding allows. For example, encoding the two values {0, 1} with a Hamming distance of 2 requires 2 bits, resulting in the valid encoding {00, 11}, and two invalid (erroneous) encodings {01, 10}. This example is identical to treating the first bit as the data bit and the second bit as an even parity bit. Note that a Hamming distance of 2 provides only the detection of a single error by noting an invalid encoding like 01 or 10, but it cannot correct it.
